Una nota su operazioni associative, trasformate integrali e problemi di caratterizzazione in statistica

Abstract

In questa nota si studia una equazione funzionale che ha interessanti applicazioni al Calcolo delle Probabilità. Si illustrano due possibilità applicative:

• (i) la scelta del nucleo delle trasformate integrali;

• (ii) la caratterizzazione di opportune classi di variabili casuali.

Abstract

If a functionf is used to probe the distribution ofX+Y for independent random variablesX andY, it is convenient to determine the expectation off(X+Y) from expectations off(X) andf(Y) separately. Formally, it is necessary to consider those functionsf for which there exists a functionT such that:

$$E(f(X + Y)) = T(E(f(X)),E(f(Y)))$$

((1))

for all independent random variablesX andY defined onA⊑R.

This work purports to study the functions ψ(X, Y) for which equation (1) is satisfied. Therefore we determine the functionsf continuous and strictly monotonic and ψ for which there exists a convenient functionT such that:

$$E(f(\psi (X,Y))) = T(E(f(X)),E(f(Y)))$$

((2))

for all independent random variablesX andY defined onA⊑R.

Equation (2) has solutions if and only if the operations ψ(x, y) admit the representation

$$\psi (x,y) = h_0 ^{ - 1} (h_0 (x) + h_0 (y))\forall x,y \in A \subseteq R$$

whereh ₀ (x) is a continuous and strictly monotonic function defined onA, closed in range with respect to the addition operation.

The solution of the problem is given by the following functions (under some suitable conditions)

$$\begin{array}{*{20}c} {f(x) = ch_0 (x) + T_{00} withT_{00} arbitraryconstant} \\ {f(x) = a + b\exp (ch_0 (x))withb \ne 0} \\ \end{array}$$

For eachf solving equation (2),T is respectively:

$$\begin{array}{*{20}c} {T(x,y) = x + y + T_{00} } \\ {T(x,y) = a + (x - a)(y - a)/bwitha = - T_{10} /T_{11} ,b = 1/T_{11} } \\ \end{array}$$

The functional equation (2) is used to determine the kernel of some integral transforms and to characterize exponential, Pareto and Weibull distributions